$A$ body starts from rest and moves with uniform acceleration. If the distance travelled by it in the first $2 \,s$ is $x_1$ and in the next $2 \,s$ is $x_2$, then $x_1$ and $x_2$ are related as

$A$ small block slides down on a smooth inclined plane,starting from rest at time $t=0$. Let $S_{n}$ be the distance travelled by the block in the interval $t=n-1$ to $t=n$. Then,the ratio $\frac{S_{n}}{S_{n+1}}$ is

The acceleration $a$ in $m/s^2$ of a particle is given by $a = 3t^2 + 2t + 2$,where $t$ is the time in seconds. If the particle starts with an initial velocity $v = 2\,m/s$ at $t = 0$,then the velocity at the end of $t = 2\,s$ is $............m/s$.

$A$ student is at a distance of $16 \ m$ from a bus when the bus begins to move with a constant acceleration of $9 \ m \ s^{-2}$. The minimum velocity with which the student should run towards the bus so as to catch it is $\alpha \sqrt{2} \ m \ s^{-1}$. The value of $\alpha$ is

The displacement $x$ (in $m$) of a particle of mass $m$ (in $kg$) moving in one dimension under the action of a force,is related to time $t$ (in $s$) by $t = \sqrt{x} + 3$. The displacement of the particle when its velocity is zero,will be

The motion of a body is given by the equation $\frac{dv(t)}{dt} = 6.0 - 3v(t)$,where $v(t)$ is speed in $m/s$ and $t$ is in $s$. If the body was at rest at $t = 0$,which of the following statements is correct?